Height: Difference between revisions

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# H(''q''''n'') ≥ H(''q'') for any non-negative integer ''n''.

If we have a function F~~(''x'')~~ which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation:

If we have a function F which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation:

<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>

A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''p'' and ''q'' are considered equivalent if the following is true:

A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''1 and ''q''2 are considered equivalent if the following is true:

<math>\displaystyle 2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q</math>

<math>\displaystyle 2^{-v_2 \left( {q_1} \right)} q_1 = 2^{-v_2 \left( {q_2} \right)} q_2</math>

where v''p''(''q'') is the [[Wikipedia: P-adic valuation|''p''-adic valuation]] of ''q''.

Or equivalently, if ''n'' has any integer solutions:

<math>\displaystyle p = 2^n q</math>

<math>\displaystyle q_1 = 2^n q_2</math>

If the above condition is met, we may then establish the following equivalence relation:

<math>\displaystyle p \equiv q</math>

<math>\displaystyle q_1 \equiv q_2</math>

By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.

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| [[Benedetti height]] (or [[Tenney height]])

| Height

| <math>~~n d~~</math>

| <math>nd</math>

| <math>2^{\large{\|q\|_{T1}}}</math>

| <math>\|q\|_{T1}</math>

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|}

Where ||''q''||T1 is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v''p''(''x'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of x.

Where ||''q''||T1 is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v''p''(''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''.

The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L1 norm on monzos, but where each prime is weighted by ''p'' rather than log (''p''). This is called "Wilson's Complexity" in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>.

@@ Line 15: / Line 15: @@
 # H(''q''<sup>''n''</sup>) ≥ H(''q'') for any non-negative integer ''n''.
-If we have a function F(''x'') which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation:
+If we have a function F which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation:
 <math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>
-A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''p'' and ''q'' are considered equivalent if the following is true:
+A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:
-<math>\displaystyle 2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q</math>
+<math>\displaystyle 2^{-v_2 \left( {q_1} \right)} q_1 = 2^{-v_2 \left( {q_2} \right)} q_2</math>
+where v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic valuation|''p''-adic valuation]] of ''q''.
 Or equivalently, if ''n'' has any integer solutions:
-<math>\displaystyle p = 2^n q</math>
+<math>\displaystyle q_1 = 2^n q_2</math>
 If the above condition is met, we may then establish the following equivalence relation:
-<math>\displaystyle p \equiv q</math>
+<math>\displaystyle q_1 \equiv q_2</math>
 By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.
@@ Line 44: / Line 46: @@
 | [[Benedetti height]] <br> (or [[Tenney height]])
 | Height
-| <math>n d</math>
+| <math>nd</math>
 | <math>2^{\large{\|q\|_{T1}}}</math>
 | <math>\|q\|_{T1}</math>
@@ Line 79: / Line 81: @@
 |}
-Where ||''q''||<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''x'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of x.
+Where ||''q''||<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''.
 The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> norm on monzos, but where each prime is weighted by ''p'' rather than log (''p''). This is called "Wilson's Complexity" in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>.